The other day I suggested an alternative way of arranging the World Cup matches in soccer. Please have a look at Reflections.

The surprise is that Round 1 has been played in the alternative World Cup thanks to a spreadsheet and a healthy random function. Greece beat Uruguay surprisingly 4-0 and have taken the lead. In Round 2 they will meet Switzerland who beat Australia convincingly 3-0. In order not to repeat matches Germany was moved down one position and so on.

Round 2 will be played as soon as the weather permits. Remember there are only four rounds before the knock-out part with 2, 4, or 8 teams takes place. Strong teams like Spain and Italy started with a loss and have to play extremely well to have a chance to reach the second stage.

- Do you have time?
- I feel time has me, but shoot.
- I want you to play a solitaire game several times in a row.
- What could me more fun?
- I will shuffle this deck of cards and put the pile face down on the table.
- I can dig that.
- You will turn one of the cards at the time starting from the top.
- OK.
- Before you do I want you to guess when you will draw the first red ace and when you will draw the second red ace.
- I have reason to believe that I will draw the first before the second.
- Brilliant! But can you be more precise? Where in the stack will the first red ace most likely be and where will the second be?
- I may have to peep?

How do you determine the best of 32 soccer teams? That’s the question.

One way is being tried out in South Africa right now. Seed some teams and make up 8 groups with 4 teams in each. Call it round 1. Let every team meet every other team in the group in round 1 and let the two best teams advance to the second round. In the second round the 16 teams play an elimination tournament, one loss and you are out.

In my opinion the system has several serious flaws.

Not every match counts. You may lose a match in round 1, but if you advance to round 2 it is of no importance.

Because of point 1 the early matches are less exciting as they may be of no importance.

The teams do not play the same number of matches. If you don’t go to round 2 you play only 3 matches, while the finalists play 7.

Some teams are seeded based on previous performance giving them an advantage. All teams should enter the tournament on the same footing.

The system only determines the best team. The losing finalists may not be the second best team and for the other teams they don’t get a final score showing how well they did.

The good news is that all of these flaws can be rectified if a modified Swiss System is used. The system goes like this:

In the first round who meets who is decided randomly.

In all other rounds you meet a team you haven’t met before that has the same number of points as you (3 points for a win, 1 for a draw, and 0 for a loss) and the same goal average.

For the cases where the rule in 2 does not uniquely determine who meets who one draws the opponents.

Four rounds are played which gives a total of 64 matches. In the present system 63 matches (8 * 6 + 8 + 4 + 2 + 1) are played.

If some teams end up with the same number of points they play an elimination tournament to determine the winner. That way you have a final match where everything is decided.

If the number of teams at the top after round 4 is not 2, 4, or 8 add the next teams to join them in round 2.

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- The problem yesterday was unfair.
- I am sorry. In what way?
- You said Euclid had proven something he hadn’t!
- Did I? Oh well, something I get things mixed up.
- Is that your only excuse?!
- Have you heard of Imre Lakatos?
- I am not quite sure. Did he write ‘How to effect children’s affections in rural Albania’?
- No, he wrote ‘Proofs and refutations: the logic of mathematical discovery’.
- I was not even close. Was he Albanian by any chance?
- He was Hungarian. More importantly, he suggested textbooks should be written in heuristic style to show how problems are solved and mathematics created.
- So the textbooks should be less authoritarian?
- Exactly, if something is true or not is for the student to ponder, not for the textbook to impose.
- Sounds like an impractical approach.
- Anyway, here is a problem Ravi a reader in India posted in a comment in New York Times the other day.
- Is it inspired by Lakatos?
- Give me a prime number.
- How about 11?
- What is 11 in base 3?
- You want me to write 11 as the sum of some 9s, 3s, and 1s?
- OK.
- 11 = 1*9 + 0*3 + 2*1. So 11 is 102 in base 3.
- Add 11 and 102.
- 113.
- Which is prime!
- So it is.
- Find the biggest prime that when added to its base 3 representation is a prime.
- What if there are infinitely many of them?
- Hadn’t thought of that!

- What are the numbers above?
- Some prime numbers I found at Wikipedia.
- That reminds me. I got a letter from a student of mine.
- A maths student?
- No she was more into art. Anyway, she has found a proof that amazes me.
- I am intrigued!
- She studied p1 * p2 * p3 * … * pn + 1 where pi is the ith prime number.
- I am lost!
- Let me give you an example. For n = 3 she multiplied the first three prime numbers and added one.
- What on earth for?
- 2 * 3 * 5 + 1 = 31.
- 31 is on the Wikipedia list.
- That is what she noticed! She proved that p1 * p2 * p3 * … * pn + 1 always gives a prime.
- I have some bad news. Euclid, who was not born yesterday, used that fact to show that there are an infinite number of primes.
- Really? I guess I should get out more often.